"Please use other door" signs are common. But would you ever say this? Or would you say "please use the other door"?
No, you would not normally say that. Signs are often expressed in a short and unspecific way. By making the text short, you can use a smaller sign, or use larger text on the same space.
The sign would have different meanings depending on the alternatives. Either:
Please use the other door.
Please use any of the other doors.
By being unspecific, the sign can be used for both situations. Also, the text is half as long as the one you would need to use if there are more than one other door, saving valuable space.
Real estate is a precious commodity.
The less you have to print on a sign, the smaller you can make it and less you have to spend on ink/paint/materials.
But to answer your question; no, I wouldn't say that. I would read the sign as "Please use the other door."
Both are acceptable and understood.
However, for me, 'Please use other door' when spoken sounds curt and I would look at the person who said it to see if they were being antagonising.
'Please use the other door' sounds more polite.
That's exactly it: "Real estate is a precious commodity."
As a result, a series of conventions has grown up surrounding signage, headlines, proofreading symbols, and other places that I cannot think of right now, in which space is tight or there is a need to be understood quickly. The conventions include abbreviations and omissions. Particularly, articles ("the," "a/an") are standard omissions in these situations. English is usually understandable and often clear without them.
I disagree that "Please use other door," when spoken sounds curt. It doesn't sound rude, it is just stilted and incorrect grammar. Written in any sort of prose, or spoken, the article is of course grammatically necessary.
I think this is an example of what Messrs H.W. & F.G. Fowler described as Headline English — done for brevity and conciseness. As OghmaOsiris said, because Real estate is a precious commodity on a sign, just as in a newspaper headline.